The BINGO telescope was designed to measure the fluctuations of the 21-cm radiation arising from the hyperfine transition of neutral hydrogen and aims to measure the Baryon Acoustic Oscillations (BAO) from such fluctuations, therefore serving as a pathfinder to future deeper intensity mapping surveys. The requirements for the Phase 1 of the projects consider a large reflector system (two 40 m-class dishes in a crossed-Dragone configuration), illuminating a focal plane with 28 horns to measure the sky with two circular polarisations in a drift scan mode to produce measurements of the radiation in intensity as well as the circular polarisation. In this paper we present the optical design for the instrument. We describe the intensity and polarisation properties of the beams and the optical arrangement of the horns in the focal plane to produce a homogeneous and well-sampled map after the end of Phase 1. Our analysis provides an optimal model for the location of the horns in the focal plane, producing a homogeneous and Nyquist sampled map after the nominal survey time. We arrive at an optimal configuration for the optical system, including the focal plane positioning and the beam behavior of the instrument. We present an estimate of the expected side lobes both for intensity and polarisation, as well as the effect of band averaging on the final side lobes. The cross polarisation leakage values for the final configuration allow us to conclude that the optical arrangement meets the requirements of the project. We conclude that the chosen optical design meets the requirements for the project in terms of polarisation purity, area coverage as well as homogeneity of coverage so that BINGO can perform a successful BAO experiment. We further conclude that the requirements on the placement and r.m.s. error on the mirrors are also achievable so that a successful experiment can be conducted.(Abridged)
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